[evoluti1]

To find the instantaneous acceleration from a standstill is relatively straightforward. If you know the mass of the robot, all you need to know is the force the drivetrain can exert. Then you can use Newton's 2nd Law to calculate acceleration. In FRC, robots are nearly always geared so that the force required to stall the motors is greater than the maximum friction force the wheels can exert against the ground. So for the instantaneous case, the force we care about is just the weight of the robot times the coefficient of friction between the wheel and the ground. (Actually, this approach is valid not just for the instantaneous case, up to but up to whatever speed corresponds to a torque on the motors' performance curve that is insufficient to "slip the wheels.")

As others have said, it gets more complicated if you want to understand the robot's motion as it accelerates. The difficulty comes from the fact that as the robot's speed increases, the amount of torque the motors can provide decreases, and so does acceleration. In other words, the available torque from the motor depends linearly on the robot's velocity, so an analytical solution would involve a differential equation.

Solving that problem would be enough of a leap from high school math, but it gets worse. There are inefficiencies in any gearbox, between the wheels and the carpet (or whatever the playing surface is), and losses in the electrical system, all of which may depend non-linearly on velocity, making the differential equation that needs to be solved more complex.

Instead, rather than solve this analytically, it may make more sense to simulate what happens as the robot accelerates. If this interests you, you should absolutely pursue it - I know I've seen white papers on Chief Delphi estimating acceleration in a "drag race." If you understand a little kinematics and how to read a motor's performance curve, you might be able to set up something quick-and-dirty in Excel and get decent results, at least for making qualitative comparisons between possible reduction options.

While I don't want to scare you away from doing the math, another way would be to load up a test chassis to 150 or so pounds, and then just swap out gear ratios and use a stopwatch. While possibly more expensive, that approach does have the advantage of being more empirical, eliminating any guesswork about the non-linear or unknown factors described above.